Determinants · Lesson 1 of 6
Understanding and Evaluating Determinants
“Turn a square array into a number and learn to evaluate determinants with a clear choice of row or column.”
• Distinguish a square matrix from its determinant and read determinant notation correctly. • Evaluate determinants of orders one, two, and three. • Explain the deletion and sign steps in a row or column expansion. • Choose an efficient expansion and check the result using a different one. • Use determinant equations and the scaling relationship |kA| = kⁿ|A|.
A number that tells us something about a matrix
A matrix records several numbers in a fixed arrangement. A determinant takes the entries of a square matrix and combines them into one number. That number will help us answer questions about equations, area, and inverses, so we begin by understanding its meaning and calculation.
Consider a₁x + b₁y = c₁ and a₂x + b₂y = c₂. Their coefficient matrix is A = [a₁, b₁; a₂, b₂]. The expression a₁b₂ − a₂b₁ decides whether the two equations have a unique solution: a nonzero value guarantees one. It depends only on the coefficients on the left, not on c₁ and c₂. This is the determinant of A. We will justify the full solution method later; at present it gives us a reason to study this particular combination.
A determinant is a number associated with a square matrix. For a matrix A it is written det A, |A|, or Δ. In this chapter we calculate determinants of orders one, two, and three with real entries.
The distinction between the array and its value matters. A = [2, 4; −1, 2] is a matrix with four entries, whereas det A is one number. Only square matrices have determinants in the sense used here: a 2 × 3 array does not have one. The notation |A| means the determinant of the matrix A; it does not mean the absolute value of every entry or the modulus of A.
In a compact array such as [2, 4; −1, 2], a semicolon separates rows. Thus the first row is 2, 4 and the second is −1, 2. A vertical-bar determinant displayed in a formula is the number associated with that square array.
We can view the determinant as a function: input a square matrix and receive its uniquely determined numerical value. Different matrices can have the same determinant, just as different inputs to other functions can have the same output. A determinant therefore tells us useful information, but it does not reconstruct all the original entries.
Determinants are used in engineering, science, economics, and social sciences wherever square coefficient arrays arise. More generally their entries and values can also be complex numbers, but all calculations in this chapter use real entries. Our concrete applications will be coordinate geometry, matrix inverses, and linear systems.
Starting with orders one and two
The order of a determinant is the number of rows, which is also the number of columns. The smallest cases let us see how a value is built from entries. In order two, the subtraction between two diagonal products is essential.
For instance, the determinant of the order-one matrix [−5] is −5, not 5. The outer bars are determinant notation. In order two, we keep brackets around negative entries during multiplication: ad − bc is a difference of products, and either product may itself be negative.
Problem
Evaluate det [2, 4; −1, 2].
- 1.The main-diagonal product is 2 × 2 = 4.
- 2.The other diagonal product is 4 × (−1) = −4.
- 3.Subtract the second product: det A = 4 − (−4) = 8. The minus in the determinant rule and the minus in the entry have different jobs.
Problem
Evaluate det [x, x+1; x−1, x].
- 1.Apply the same order-two rule before substituting any value for x: Δ = x·x − (x+1)(x−1).
- 2.The product (x+1)(x−1) is x²−1, so Δ = x² − (x²−1).
- 3.Distribute the subtraction to get x² − x² + 1 = 1. Although all four entries vary with x, the determinant is always 1.
Equal matrices have equal determinants, but equal determinants do not force the matrices to be equal. For example, I₂ and [2,0;0,1/2] both have determinant 1. Their entries are different.
Building an order-three expansion
An order-three determinant can be reduced to three order-two determinants. Choose a row or column, take each entry in it, delete the entry’s entire row and column, and multiply by the determinant of the remaining square. A sign determined by the entry’s position completes each contribution.
Let A = [aᵢⱼ] have order three. The subscripts i and j give the row and column of an entry. In the first row, a₁₁ leaves rows 2 and 3 and columns 2 and 3 after deletion. The entry a₁₂ leaves rows 2 and 3 but columns 1 and 3. The entry a₁₃ leaves rows 2 and 3 and columns 1 and 2. Keep the surviving entries in their original order.
More generally, the multiplier for a position (i,j) is (−1)^(i+j). An even value of i+j gives +1; an odd value gives −1. This is a position sign, independent of the sign of the entry. A negative entry in a minus position can therefore produce a positive multiplier. The pattern is +,−,+ in row 1; −,+,− in row 2; and +,−,+ in row 3.
Evaluating the three small determinants in the first-row expansion gives a₁₁(a₂₂a₃₃−a₂₃a₃₂) − a₁₂(a₂₁a₃₃−a₂₃a₃₁) + a₁₃(a₂₁a₃₂−a₂₂a₃₁). This expression is the full order-three calculation. There is no multiplication of three corresponding entries alone: all three signed contributions are needed unless some are zero.
Why the choice of row or column is free
There are six possible expansions for an order-three determinant: three rows and three columns. They give the same result. To understand that freedom, compare the algebra behind different choices rather than memorising a separate procedure for each one.
Distribute each bracket in these expressions. The same six products appear: +a₁₁a₂₂a₃₃, −a₁₁a₂₃a₃₂, −a₁₂a₂₁a₃₃, +a₁₂a₂₃a₃₁, +a₁₃a₂₁a₃₂, and −a₁₃a₂₂a₃₁. Only the order in which they are written changes. Expansions along row 3 and columns 2 and 3 can be checked in the same way. This explains why we may choose the easiest row or column without changing the determinant.
Problem
Evaluate det [1,2,4; −1,3,0; 4,1,0].
- 1.Column 3 has only one nonzero entry, 4 at position (1,3), whose position sign is positive.
- 2.Delete row 1 and column 3. The remaining determinant is det [−1,3;4,1] = (−1)·1 − 3·4 = −13.
- 3.The other two column contributions are zero because their entries are zero. Thus Δ = 4(−13) = −52.
- 4.Choosing a column with zeros reduces arithmetic; it does not permit us to ignore nonzero terms in another chosen row.
Problem
Evaluate det [0,sin α,−cos α; −sin α,0,sin β; cos α,−sin β,0].
- 1.Expand along row 1. The first contribution is zero because a₁₁ = 0.
- 2.The middle contribution is −sin α times det [−sin α,sin β;cos α,0] = −sin α(−sin β cos α).
- 3.The last contribution is (−cos α) times det [−sin α,0;cos α,−sin β] = −cos α(sin α sin β).
- 4.The surviving terms are sin α sin β cos α − cos α sin α sin β = 0. Their cancellation is valid for all real α and β.
Determinant equations and scaling
Once a determinant has been evaluated, it is an ordinary number or algebraic expression. We can equate two determinant values and solve for an unknown. Scaling an entire matrix is another useful pattern, but its effect on the determinant depends on the order.
Problem
Find x if det [3,x;x,1] = det [3,2;4,1].
- 1.The left determinant is 3−x². The right determinant is 3−8 = −5.
- 2.Set the values equal: 3−x² = −5, so x² = 8.
- 3.Both signs satisfy this equation: x = 2√2 or x = −2√2. Checking either value in 3−x² gives −5.
For order two, replacing every entry of B = [a,b;c,d] by k times that entry gives det(kB) = (ka)(kd) − (kb)(kc) = k²(ad−bc). For order three, each of the six expanded products contains three entries, so each gains three factors of k. The order-one case gains one factor. Thus the power counts the order of the square matrix, rather than the number of entries.
Problem
For B = [1,1;2,0] and A = 2B, compare their determinants.
- 1.First, det B = 1·0 − 1·2 = −2.
- 2.The scaled matrix is A = [2,2;4,0], so det A = 2·0 − 2·4 = −8.
- 3.Since the order is two, 2² det B = 4(−2) = −8. The scaling factor for the determinant is 4, not 2.
Early Chinese calculating-board arrangements of coefficients supported elimination by rows and columns. Seki Kowa used related ideas in seventeenth-century Japan. Vandermonde recognised determinants as functions in their own right; Laplace developed expansion methods, while Lagrange and Gauss used determinants in further mathematics. Binet and Cauchy studied product relationships, and Jacobi made major contributions to the theory. These developments explain the chapter’s two themes: a determinant is a function of a square array, and it helps analyse equations.
Quiz
Which statement correctly distinguishes A and det A?
What is det [−7], for a matrix of order one?
Evaluate det [3,−2;4,1].
Which position has a negative sign in a cofactor-style expansion?
A matrix B has order three and det B = −2. What is det(3B)?
An order-three matrix has a column containing two zeros. Why expand along that column?
If det [x,0;0,x] = 9, what are the possible real x values?
Practice Problems
- Evaluate det [2,4;−5,−1], keeping the subtraction and the negative products separate.
- Evaluate det [cos θ,−sin θ;sin θ,cos θ] and det [x²−x+1,x−1;x+1,x+1]. State the identity or factorisation you use.
- For A = [1,2;4,2], calculate det A and det(2A) directly, then verify the scaling relationship.
- For A = [1,0,1;0,1,2;0,0,4], calculate det A and det(3A), and explain the power 27.
- Evaluate each determinant, selecting a useful row or column: [3,−1,−2;0,0,−1;3,−5,0], [3,−4,5;1,1,−2;2,3,1], [0,1,2;−1,0,−3;−2,3,0], and [2,−1,−2;0,2,−1;3,−5,0].
- Find det A for A = [1,1,−2;2,1,−3;5,4,−9]. Verify the value with a second expansion.
- Solve det [2,4;5,1] = det [2x,4;6,x], then solve det [2,3;4,5] = det [x,3;2x,5].
- Find all real x satisfying det [x,2;18,x] = det [6,2;18,6]. Explain why equality of determinants does not require x to equal 6.
- For a symbolic order-three matrix, expand along row 3, column 2, and column 3. Distribute the terms and compare their six signed products with the first-row expression.
Key Takeaways
• A determinant is one number associated with a square matrix; its bars do not mean modulus. • Order-two determinants are a difference of two products, with signs preserved carefully. • An order-three expansion uses an entry, its position sign, and the determinant left after deleting its row and column. • All row and column expansions agree; zeros help us choose an efficient one. • Scaling every entry by k scales the determinant by kⁿ, where n is the order. • Determinant equations may have more than one solution, and equal determinants need not come from equal matrices.
Previous
Start of chapter
Next · Lesson 2
Determinants and the Area of a Triangle